Version 2.0 - Major revision (November 2025) We construct a framework deriving quantum uncertainty from information-theoretic axioms without presupposing quantum mechanics, Planck’s constant, or commutation relations. From four axioms—finite information in causally-connected regions, relational specification costs, subsystem decomposability, and incompatibility penalties—we prove that conjugate observables satisfy ∆X · ∆P ≥ S0 where S0 is a universal action scale. Consistency with holography requires surface (not volume) scaling of information capacity, I(R) ∼ Rd−1/ℓd−1 ∗ . Matching black hole entropy uniquely identifies ℓ∗ = ℓP (Planck length). Thermodynamic arguments prove S0 is universal across all systems; empirical measurements identify S0 = ℏ. For gravitationally-bound systems, we derive Rreq/Ravail = rS/R from the Bekenstein bound, predicting enhanced uncertainty near compact objects: 12% for neutron stars (marginally detectable with LIGO O5 stacking analysis), 50% for extreme mass-ratio inspirals (testable with LISA post-2035). We demonstrate why white dwarfs (rS/R ∼ 10−4) show no observable effects and explain that the GW170817 gamma-ray delay has an astrophysical origin unrelated to graviton dispersion. Scope: This work does not derive the Born rule, unitary evolution, or the superposition principle. Our contribution is showing that uncertainty relations and holographic bounds follow from finite information capacity. CHANGES FROM v1.0: - Added Axiom 4 (Incompatibility Penalty) to rigorously derive uncertainty relation - Corrected derivation of saturation parameter r_S/R from Bekenstein bound - Removed numerical ambiguity in Planck length identification (π factor) - Realistic assessment of observational feasibility (white dwarfs: not detectable) - Clarified that GW170817 gamma-ray delay is astrophysical, not graviton dispersion - Removed speculative philosophical content - Strengthened holographic scaling derivation
Version 2.0 - Major revision (November 2025) We construct a framework deriving quantum uncertainty from information-theoretic axioms without presupposing quantum mechanics, Planck’s constant, or commutation relations. From four axioms—finite information in causally-connected regions, relational specification costs, subsystem decomposability, and incompatibility penalties—we prove that conjugate observables satisfy ∆X · ∆P ≥ S0 where S0 is a universal action scale. Consistency with holography requires surface (not volume) scaling of information capacity, I(R) ∼ Rd−1/ℓd−1 ∗ . Matching black hole entropy uniquely identifies ℓ∗ = ℓP (Planck length). Thermodynamic arguments prove S0 is universal across all systems; empirical measurements identify S0 = ℏ. For gravitationally-bound systems, we derive Rreq/Ravail = rS/R from the Bekenstein bound, predicting enhanced uncertainty near compact objects: 12% for neutron stars (marginally detectable with LIGO O5 stacking analysis), 50% for extreme mass-ratio inspirals (testable with LISA post-2035). We demonstrate why white dwarfs (rS/R ∼ 10−4) show no observable effects and explain that the GW170817 gamma-ray delay has an astrophysical origin unrelated to graviton dispersion. Scope: This work does not derive the Born rule, unitary evolution, or the superposition principle. Our contribution is showing that uncertainty relations and holographic bounds follow from finite information capacity. CHANGES FROM v1.0: - Added Axiom 4 (Incompatibility Penalty) to rigorously derive uncertainty relation - Corrected derivation of saturation parameter r_S/R from Bekenstein bound - Removed numerical ambiguity in Planck length identification (π factor) - Realistic assessment of observational feasibility (white dwarfs: not detectable) - Clarified that GW170817 gamma-ray delay is astrophysical, not graviton dispersion - Removed speculative philosophical content - Strengthened holographic scaling derivation
The Problem of Time — the apparent incompatibility between quantum mechanics' time parameter t and general relativity's proper time τ — has been treated as a fundamental obstacle to quantum gravity. I argue that this is a level-confusion: both theories define time within their own entity-theoretic frameworks, generating an apparent opposition that disappears when time is relocated to a meta-structural layer. Drawing on the meta-level temporal structure T_SDT := (V_G, ≺) (Zeng 2026), the paper formalizes QM and GR time as two projections of a single Universal Time Formula, parameterized by entity-theoretic tokens. Four token specializations recover the corresponding time concepts. A Bridge Theorem establishes mutual compatibility of all token projections. A bidirectional QM-GR channel is constructed (forward: GR geometry → QM phase; reverse: discrete chain-pair → Lorentzian spacetime), composing into a self-consistent closed loop. Three further results move beyond dissolution toward unification. First, a Madelung-bridge construction derives the Schrödinger equation — Hilbert space, unitarity, Hamiltonian, quantum potential, Born rule — from the determination-graph axioms together with the continuum embedding (CL1), with no independent QM input; the derivation extends to Klein-Gordon. Second, a Planck-scale token μ = t_P / |A_{v'}| is constructed on the discrete 𝒢_D, where |A_{v'}| is the admissibility cardinality; under CL1, this single step-measure recovers QM (|A| ≫ 1) and GR (|A| = 1) as its two continuum limits, with Einstein's equation proved in the classical regime (|A| = 1) via a two-stage argument (Benincasa-Dowker action convergence + standard scalar-tensor variation). The exact quantum-gravitational correction is derived as a scalar-tensor equation with φ = 1/|A|. Third, the discrete determination graph without continuum embedding constitutes a self-contained Planck-scale quantum gravity theory: it has a well-defined path integral, a finite-dimensional S-matrix (unitary in the κ = 0 sector, isometric in the κ > 0 sector; no UV divergences), preserves information across black hole horizons, resolves singularities, derives the Past Hypothesis, and provides structural origins for gauge symmetry, Hawking radiation, spectral dimensionality, and the spin-statistics connection. The Problem of Time is not merely dissolved — it is replaced by a constructive framework in which QM and GR are unified as two limits of a single discrete structure.
This work presents an extended, mathematically complete formulation of the Information Geometry Unification Principle (IGUP), a foundational framework in which quantum mechanics, statistical thermodynamics, general relativity, and spacetime geometry emerge from a single information-theoretic ontology. We begin with rigorous axioms defining information as the primary physical substance, then systematically construct the Bures–Fisher–Rao metric on the manifold of quantum density matrices. We derive the universal information action functional and perform full variational derivations to obtain the unified field equations. We show explicitly how standard physical theories including the Schrödinger equation, Heisenberg picture, Liouville mechanics, classical thermodynamics, and Einstein field equations are recovered in appropriate limiting regimes. We further derive geometric expressions for entanglement, entropy, temperature, energy, and force. The theory resolves longstanding conceptual tensions between unitary quantum evolution and gravitational irreversibility, provides a natural resolution to the black hole information paradox, and yields eight concrete, experimentally testable predictions. This extended version includes full tensor calculus, covariant derivatives, curvature expansions, and limit theorems, amounting to a self-contained theory of everything based on Geometry=Information.
We propose that uncertainty relations and entanglement are not limited to the handful of conjugate pairs discovered in quantum mechanics, but represent a universal principle governing all boundary creation in reality. Reality is born when perfect symmetry is broken by exactly one physical inequality-any minimal, thermodynamically paid bias between entangled states. There is no privileged numeric ratio; the relevant invariant is the existence of a nonzero inequality Δ ≠ 0 that survives because it is amplified across N ≥ Nc degrees of freedom and pays at least the Landauer cost kB T ln 2 per bit. This framework predicts infinite complementary pairs across all domains-physical, biological, informational, and conscious-each imposing measurement limits and correlation structures. We derive a universal uncertainty formula ΔP × ΔP' ≥ ℏ_domain/2 applicable to any measurable property and its complement, where ℏ_domain represents the fundamental quantum of action for that domain. This theory explains why we've discovered only a small subset of possible uncertainty relations and predicts numerous testable new pairs including Information- Entropy, Growth-Stability, Causality-Correlation, and Self-Other boundaries.
Standard quantum interpretations-based on point trajectories, branching wave functions, or instantaneous projection-rely on two continuum idealizations: infinite spatial information density and infinite measurement bandwidth. A central foundational question is whether complex Hilbert-space kinematics, noncommuting observables, and single-outcome collapse follow from real physical fields under finite information capacity and bandwidth. We formulate Objective Phase-Space Quantum Information-Capacity Mechanics (QICM), specifying the state on a configuration manifold by a real probability density and a compact flow potential. Three axioms govern the theory: the first requires finite Fisher information and defines the potential on a flat circle bundle with quantized holonomy and half-period nodal jumps, resolving Wallstrom's objection; the second sets an elementary resolution quantum from the Bloch-sphere Fisher solid angle; and the third pairs Fisher localization energy with a Nyquist-Planck capacity bound in stochastic equations for closed flow and continuous measurement. On the reduced tangent space, the symplectic form, Fisher-Rao metric, and complex structure form a Kahler geometry inducing the Hermitian inner product, self-adjoint operators, and Heisenberg commutators. The axioms yield uncertainty bounds, the minimum resolution time, oscillator and hydrogen spectra, martingale Born-rule collapse, and the Lindblad equation, and extend to relativistic fields, half-integer spin, Pauli pressure, Compton-scale ultraviolet regularization, and ensemble no-signaling. QICM thus unifies unitary evolution and collapse in a single-world field dynamics without superluminal signaling, yielding testable bounds on compression times, readout horizons, and short-pulse thresholds in circuit-QED, optomechanics, and trapped ions.
ONE AXIOM BRIDGES: Possibility Selection Principle — Formalization, Collapse, and Born Rule Description: This document provides the formal mathematical bridge between the pure algebraic foundations of the ONE AXIOM project and its physical applications. It presents a rigorous derivation of the fundamental laws of physics starting from a single axiomatic symmetry group: G = S4 x Z2^3. Key Scientific Contributions: Derivation of the Born Rule (p=2): Proves that quantum probability is not a postulate but a theorem. The exponent p=2 is shown to be the unique solution emerging from the OCR-OER adjunction structure. Quantum Collapse as Domain Update: Formalizes "collapse" as a relational update of the Schreier graph domain (Gamma to Gamma_A) rather than a physical process, resolving long-standing paradoxes of "spooky action at a distance". Emergence of Spacetime and Gravity: Demonstrates the construction of 4D spacetime (M4) from group representation theory. Einstein's field equations are derived via two independent, proven paths: Jacobson's thermodynamic approach and Ollivier-Ricci curvature on the Schreier graph. Resolution of the Cosmological Constant Problem: Provides a natural derivation of the cosmological constant Lambda ~ 10^-122 from the "coherence floor" of the universe, eliminating the need for fine-tuning. Bekenstein-Hawking Entropy: Derives the coefficient lambda = 1/4 through triple-verified categorical and structural paths. Methodology: The work employs a "Dual-Track Derivation" methodology, requiring convergence between Ontological [O] (structural) and Epistemological [E] (computational) paths before a result is considered proven. Status: This version marks the completion of the foundational triad of the ONE AXIOM framework (0C, 0A and 0B documents)
We extend the Informational Gravity framework (Papers I-II) to quantum mechanics. Papers I and II established the nodes of the causal network (informationally bound energy) and defined their spatial relationships through bidirectional causal connectivity. This paper identifies the connections: massless interactions as timeless field potentials that link nodes without temporal extension. This identification yields a taxonomy of physical relationships - causal connections, entanglement, and the absence of connection - that proposes to reframe the EPR paradox without invoking nonlocality, accommodates Bell inequality violations within its ontological categories, provides an account of delayed choice experiments, and aims to dissolve the measurement problem by proposing that every interaction between a timeless connection and a time-bound system (any system with mass and proper time) constitutes a projection - removing the need for a special category of "measurement." An independent result - that finite informational capacity of a discrete network grounds a structural reading of the uncertainty principle - connects quantum indeterminacy to the framework's foundational structure. The framework does not modify quantum mechanics but offers an interpretive ontology in which its features become natural consequences of a causal network rather than irreducible axioms. We argue that this ontology renders the Many Worlds interpretation unnecessary.